Click
Here
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, C09011, doi:10.1029/2006JC004024, 2007
for
Full
Article
On the extreme statistics of long-crested deep
water waves: Theory and experiments
Nobuhito Mori,1 Miguel Onorato,2 Peter A. E. M. Janssen,3 Alfred R. Osborne,2
and Marina Serio2
Received 21 November 2006; revised 14 March 2007; accepted 7 June 2007; published 15 September 2007.
[1] Quasi-resonant four-wave interactions may influence the statistical properties of
deep water long-crested surface gravity waves. As a consequence, the wave height
exceedance probability can substantially deviate from the expected distribution obtained
by assuming that waves are linear. Here the occurrence probability of extreme events
recently derived by N. Mori and P. Janssen (2006) is compared with wave tank data, where
strong departures from Gaussian behavior are observed. Experimental wave height,
maximum wave height distribution, and probability of occurrence of freak waves are
compared with theoretical expectations. The theory well predicts extreme waves in
nonlinear wavefields.
Citation: Mori, N., M. Onorato, P. A. E. M. Janssen, A. R. Osborne, and M. Serio (2007), On the extreme statistics of long-crested
deep water waves: Theory and experiments, J. Geophys. Res., 112, C09011, doi:10.1029/2006JC004024.
1. Introduction
[2] In the last decades a lot of effort in the oceanographic
community has been dedicated to the study of extreme
events in the ocean. In years 2004 and 2005, three conferences/workshop, Rogue Waves 2004 held in Brest, France
[Olagnon and Prevosto, 2004], Rogue Waves in Hawaii
[Muller and Henderson, 2005], and ICMS Workshop on
Rogue Waves in Edinburgh, UK, and sessions in large
conferences, such as European Geosciences Union
[Pelinovsky and Kharif, 2006], were completely devoted
to the study of extreme waves. The state on the art on the
subject can be found in the proceedings of the conferences.
Nowadays, it is accepted that at least four mechanisms are
responsible for the formation of extreme waves. The first
one is just a linear superposition of waves; in this case the
probability distribution for wave height in the limit of the
narrow-band approximation obeys a Rayleigh distribution
[Longuet-Higgins, 1952]; corrections due to finite spectral
bandwidth have been obtained [Næss, 1985; Boccotti, 1989;
Tayfun, 1981]. Wave crest statistics can be achieved by
using the second-order theory developed by LonguetHiggins [1963]. In the narrow-band approximation, the
probability distribution for wave crests has been given by
Tayfun [1980] (for finite bandwidth, see Fedele and Arena
[2005]). The second mechanism is the interaction of waves
with currents: linear theory can explain the formation of
extreme waves using ray theory. The statistical properties of
the surface elevation as a function of the properties of the
currents are so far unknown. The third mechanism, the one
1
2
3
Graduate School of Engineering, Osaka City University, Osaka, Japan.
Dipartmento di Fisica Generale, Università di Torino, Turin, Italy.
European Centre for Medium-Range Weather Forecasts, Reading, UK.
Copyright 2007 by the American Geophysical Union.
0148-0227/07/2006JC004024$09.00
that will be mainly discussed here, concerns the generation
of extreme events as a result of the modulational instability,
i.e., a four wave quasi-resonant interaction process [e.g.,
Yasuda et al., 1992; Yasuda and Mori, 1994; Onorato et al.,
2001; Janssen, 2003]. This mechanism becomes relevant
for long-crested waves; in the case of waves with directional
spreading it is still not clear what is the role of the
modulational instability and consequent formation of extreme events. Numerical results in freely decaying case
[Onorato et al., 2002; Socquet-Juglard et al., 2005] have
shown that the addition of a directional spreading decreases
the probability of formation of extreme waves, leading to
wave crests distributed according to the Tayfun distribution
[Socquet-Juglard et al., 2005]. The fourth mechanism is
related to crossing sea states, i.e., two sea systems, for
example a swell and a wind sea, with different directions
that coexists in some region of the ocean. In deep water,
modulational instability in crossing seas can also play a role
[Onorato et al., 2006]: the presence of a second sea system
can trigger some instability in the first sea state that, if
alone, would be stable. The evidence of freak wave generation in the real ocean analyzing were reported by analyzing
field data of the North Sea [Stansell et al., 2003; Guedes
Soares et al., 2003], the Sea of Japan [Yasuda and Mori,
1997; Yasuda et al., 1997], and the Gulf of Mexico [Guedes
Soares et al., 2004].
[3] An ultimate goal in all these studies of extreme waves
is indeed the determination of the shape of the exceedance
probability function. Once the analytical form is found for
all sea state conditions (for example as a function of the
wave spectrum), the probability of finding a wave that
exceeds a certain height can be directly estimated. For
example, if one considers the Rayleigh distribution as a
reference distribution for wave heights then, the probability
of appearance of a wave larger than twice the significant
wave height is about 1/2980. The one difficulty of extreme
C09011
1 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
study is the verification of the theory. The statistical theories
are evaluated under the assumptions of stationarity. However, the real sea state changes both spatially and temporally. Therefore the statistical theories are difficult to verify
[i.e., Mori et al., 2002]. In addition, freak wave prediction
requires the prediction of the maximum wave height distribution. Thus the verification of the wave height distribution
is insufficient and the discussion of the maximum wave
height distribution is needed to complete the verification of
freak wave prediction [Stansell et al., 2003; Mori and
Janssen, 2006].
[4] Recently some experimental work has been conducted in a very large wave tank facility in Norway
[Onorato et al., 2004, 2005b, 2006]. One of the aims of
these experiments was to determine the shape of the distribution function for wave heights for random, long-crested
waves in deep water. The experiments have highlighted that
there are some conditions for which the exceedance probability derived from the Rayleigh distribution underestimates by almost an order of magnitude the probability of
measuring a wave larger than twice the significant wave
height. This discrepancy could not be explained in terms of
the second-order theory and has been associated to the
modulational instability phenomenon that takes place in
random waves. The presence of extreme events is accompanied by a strong departure of the surface elevation
kurtosis from the expected Gaussian value. The relationship
between the kurtosis and the extreme wave events has been
discussed during the last decade (e.g., for the laboratory data
[Mori et al., 1997], for the numerical results [Yasuda and
Mori, 1994], and for the field data [Guedes Soares et al.,
2004]). This is of course not a surprise because the kurtosis
is the fourth-order moment of the probability density
function; therefore it is a measure of the relevance of the
tails in a distribution. Mori and Janssen [2006; see also
Tayfun and Lo, 1990; Mori and Yasuda, 2002] discuss the
formal relation between the kurtosis and the exceedance
probability for wave height. The kurtosis enters in the
distribution function as a correction to the Rayleigh distribution: when the kurtosis tends to three, the expected
Gaussian values, then the distribution function tends to
the Rayleigh distribution. In this context, the changes in
the kurtosis can be evaluated once the evolution of the wave
spectrum is known [see Janssen, 2003]. Mori and Janssen
[2006] derived the wave height exceedance probability as a
function of the considered number of waves.
[5] The conditions under which large deviations have
been observed experimentally are characterized by large
steepness and narrow band spectra. This experimental result
was consistent with previous analytical and numerical
studies in weakly nonlinear wave models [Onorato et al.,
2001; Janssen, 2003] in which the ratio between the
steepness, a measure of the nonlinearity and the spectral
bandwidth, a measure of the dispersion, has been identified
as an important parameter for determining the probability of
finding a large wave. After this ratio between nonlinearity
and dispersion was named the Benjamin-Feir-Index (BFI),
its relation to the kurtosis was given by Janssen [2003] in
the limit of large times and for a narrow banded spectra. The
final result is that the kurtosis depends on the square of the
BFI. The role of the skewness in the wave height distribu-
C09011
tion is less important with respect to kurtosis. The skewness
comes usually as a result of second-order corrections
(bound modes) and is weakly affected by the dynamics of
free waves [Onorato et al., 2005a].
[6] A different approach, also based on the dynamics of
cubic nonlinearity, has been used by Fedele and Tayfun
[2006] for finding the exceedance probability for wave
height. The approach is based on an optimization problem,
requiring that at a specific space and time, the Fourier
phases are all equal, with the constraints that the conservation laws of the Zakharov equation are satisfied. Fedele and
Tayfun [2006] derived an analytical form for the exceedance
probability; nevertheless, it is still not clear if this distribution function is the result of nonlinear corrections to a quasilinear focusing process or that it really takes into account
the modulational instability as a nonlinear self-focusing
mechanism.
[7] In the present paper we will make a detailed comparison between experimental data from the Marintek wave
tank facility and theory by Mori and Janssen [2006] of a
number of statistical parameters, namely, the kurtosis evolution owing to the four wave interactions, the wave height
distribution and the maximum wave height distribution. The
paper is organized as follows: in section 2 a summary of the
results derived by Mori and Janssen [2006] is given, in
section 3 the experiment performed at Marintek is described, the comparison of experimental data and theory is
reported in section 4. The results give the universal theory
of the extreme wave generation as a consequence of four
wave interactions in a unidirectional wave train.
2. Theoretical Background
2.1. Quasi-Resonant Four Wave Interactions and
High-Order Moments
[8] As previously mentioned, the modulational instability
is a quasi-resonant interaction process, i.e., wave numbers
and frequencies satisfy the following conditions:
~
k2 ~
k3 ~
k4 ¼ 0 and
k1 þ ~
k2 w ~
k3 w ~
k4 2 ;
w ~
k1 þ w ~
ð1Þ
here is a small parameter which corresponds to the
steepness in deep water waves. More in particular, the
modulational instability takes place when two wave
k 2 and ~
k 3 and ~
k 4 are two
numbers are the same ~
k1 = ~
~
sidebands separated from k 1 by Dk, which should be small
in order to satisfy the condition (1). The standard kinetic
equation that describes the evolution of the wave spectrum
in time [Hasselmann, 1962] is formally only valid for large
times, O(4w1
0 ), and for exact resonances; its extension to
quasi-resonant interactions has been done by Janssen
[2003] where a kinetic equation, which should be valid
also on the timescale of the modulational instability, has
been derived [see also Annekov and Shrira, 2006]. If one
then considers the evolution of higher-order moments such
as the kurtosis, it turns out that the quasi-resonant
interactions are responsible for deviations from Gaussian
values. Janssen [2003] investigated the explicit relation
between nonlinear interactions of free waves and the fourth-
2 of 10
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
order moment of the surface elevation h(~
x, t) in deep water.
The result is
k40 ¼
hh4 i
12
3¼ 2 2
m20
g m0
Z
1
1 2
pð H ÞdH ¼ He8H ½1 þ k40 AH ð H ÞdH;
4
ð2Þ
where m0 is the variance of the surface elevation h, k40 is
the fourth-order cumulant of the surface elevation (equals to
kurtosis minus 3), g is the acceleration of gravity, k is the
wave number, w is the angular frequency, N is the wave
action spectral density, T1,2,3,4 is the coupling coefficient in
the Zakharov equation (see Krasitskii [1994] for its
k1 + ~
k2 ~
k3 ~
k 4),
analytical form), d1+234 = d(~
k 1 d~
k 2 d~
k 3 d~
k 4 and Rr = (1 cos(Dw t))/Dw
d~
k 1,2,3,4 = d~
is the resonant function. In the limit of large times Rr !
P/Dw, where Dw = w1 + w2 w3 w4 and P denotes the
principle value of the integral to avoid singularity in the
integral.
[9] In the narrow-band approximation, assuming that the
spectrum E(w) has a Gaussian shape:
E ðwÞ ¼
m0
1 2
pffiffiffiffiffiffi e2n ;
sw 2p
ð3Þ
where n = (w w0)/sw and sw is the spectral bandwidth, the
integral in (2) for large times becomes:
k40
242
¼ 2 P
D
Z
wave height distribution are given by Mori and Janssen
[2006]:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d~
k1;2;3;4 T1;2;3;4 w1 w2 w3 w4
d 1þ234 Rr ðDw; t ÞN1 N2 N3
dn 1;2;3
e2½n1 þn2 þn3 1
2
2
2
2
ð2pÞ3=2 ðn 1 þ n 2 n 3 Þ n 21 n 22 þ n 23
;
ð4Þ
1
where BFI is defined as by Janssen [2003]
BFI ¼
pffiffiffi
2:
D
ð6Þ
Equation (5) is the simplified prediction equation of the
kurtosis of the surface elevation assuming a narrow-band,
unidirectional wave train, but the full description requires
the evaluation of a six dimensional integral in wave number
space, equation (2).
ð7Þ
ð8Þ
pffiffiffiffiffiffi
where H is the wave height normalized by hrms = m0 ,
k40 is defined in equation (4) and AH(H) and BH(H) are
polynomials defined as
AH ð H Þ ¼
1 4
H 32H 2 þ 128 ;
384
ð9Þ
1 2 2
H H 16 :
384
ð10Þ
BH ð H Þ ¼
Note that these distributions describe the deviation from
linear statistics under the hypothesis of a narrow-band,
weakly nonlinear wave train. Second-order contributions,
which are important for the distribution of wave crests, can
be included by using a Tayfun-like approach [Tayfun,
1980; see also Tayfun, 2006].
[11] The probability distribution function, pm, and the
exceedance probability, Pm, of the maximum wave height
can also be given as a function of the fourth cumulant of the
surface elevation k40 and the number of waves N recorded
in the wave train,
pm ðHmax Þ ¼
2
Hmax
N
Hmax e 8 ½1 þ k40 AH ðHmax Þ
4
exp N e
ð5Þ
2
PH ð H Þ ¼ e8H ½1 þ k40 BH ð H Þ;
pffiffiffiffiffiffi
where = k0 m0 is the steepness parameter and D = sw /w0
is the relative spectral bandwidth. The integral can be
evaluated analytically to obtain:
p
k40 ¼ pffiffiffi BFI 2
3
C09011
2
Hmax
8
½1 þ k40 BH ðHmax Þ ;
2
Hmax
Pm ðHmax Þ ¼ 1 exp N e 8 ½1 þ k40 BH ðHmax Þ ;
ð11Þ
ð12Þ
where Hmax is the maximum wave height normalized by
hrms. Mori and Janssen’s [2006] comparison of the
theoretical wave height distribution with field data showed
qualitative agreement. For k40 = 0, results are identical to
the ones following from the Rayleigh distribution.
[12] Note that simpler looking expressions for the wave
height and maximum wave height distribution may be
obtained
byffi normalizing with the significant wave height
pffiffiffiffiffi
Hs = 4 m0 . We record these expressions for completeness.
Hence normalizing with the significant wave height, equations (7) –(12) become
2.2. Wave Height and Maximum Wave Height
Distributions
[10] In order to include nonlinear effects in the wave
height distribution function giving possible deviations from
the Rayleigh statistics, the standard approach is to use the
Edgeworth series developed at the beginning of the last
century [e.g., Edgeworth, 1907]. The theory for wave height
is described by Tayfun and Lo [1990], Mori and Yasuda
[2002], and Mori and Janssen [2006]. The resulting distribution has been named the Modified Edgeworth Rayleigh
(MER) distribution. The MER wave height and exceedance
3 of 10
p*ð H*ÞdH* ¼ 4H*e2H * ½1 þ k40 AH *ð H*ÞdH*;
2
2
PH *ð H*Þ ¼ e2H * 1 þ k40 BH*ð H*Þ ;
AH*ð H*Þ ¼
1
2H *4 4H *2 þ 1 ;
3
2
BH *ð H*Þ ¼ H *2 H *2 1 :
3
ð13Þ
ð14Þ
ð15Þ
ð16Þ
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
C09011
Figure 1. Sketch of the wave tank facility at Marintek and location of wave probes.
while the maximum wave height distribution and the
exceedance probability becomes
*2 * ¼ 4NHmax
* e2Hmax
*
p*m Hmax
1 þ k40 AH* Hmax
*2 *
exp N e2Hmax 1 þ k40 BH* Hmax
;
*2 * ¼ 1 exp N e2Hmax
*
:
Pm* Hmax
1 þ k40 BH* Hmax
ð17Þ
ð18Þ
where Hmax
* is the maximum wave height normalized by Hs.
[13] If one defines a freak wave as a wave whose height
Hfreak 2Hs, we obtain from equation (12) or (18) the
following simple formula to predict the occurrence probability of a freak wave as function of N and k40,
Pfreak ¼ 1 exp½bN ð1 þ 8k40 Þ
ð19Þ
where b = e8 is constant. The term 8k40 is the nonlinear
correction to linear theory for the maximum wave height
distribution. Thus the nonlinear correction to the maximum
wave height depends on k40.
[14] To summarize the above discussion, we can state that
the quasi-resonant four wave interactions introduce deviations from linear expectations of the statistics of surface
elevation; in particular, for weakly nonlinear, narrow-banded
and long-crested wave trains, the kurtosis evolves according
to equation (2). In the narrow-band approximation, the
kurtosis is related to the BFI; the tail of wave height
distribution depends on the kurtosis/BFI and increases as
the kurtosis increases. Finally, the maximum wave height
distribution depends on both the number of waves in the
wave train (record length) and the kurtosis; see
equation (11).
3. Experimental Data
[15] The experiment was carried out in the long-wave
flume at Marintek; experimental details and some data
analysis are given by Onorato et al. [2006]. The length of
the tank is 270 m and its width is 10.5 m; the depth is
10 meters for the first 85 meters and 5 meters for the rest of
the flume. For the wavelengths considered in the experiments, the deep water conditions apply through the tank. A
horizontally double-hinged flap type wave maker located at
one end of the tank was used to generate the waves. A
sloping beach is located at the far end of the tank opposite
the wave maker used. The wave surface elevation was
measured simultaneously by 19 probes placed at different
locations along the flume. The sampling frequency for each
probe was 40 Hz. The capacitance wave gages were used
for the surface displacement measurements. A view of the
flume with the location of the probes is shown in Figure 1.
JONSWAP random wave signals where generated at the
wave maker as sums of independent harmonic components,
by means of the inverse Fast Fourier Transform of complex
random Fourier amplitudes. These were prepared according
to the ‘‘random phase approach’’ by using random Fourier
amplitudes as well as random phases. This implies that
because of the central limit theorem, k40 should ideally be
equal to zero and m4 equal to 3. In the original experiment,
three different JONSWAP spectra with different values of
BFI have been investigated. Here we will consider only the
most interesting case, i.e., the most nonlinear case, which
corresponds to the largest BFI; this corresponds to a
JONSWAP spectrum with g = 6 and significant wave height
of 16 cm, initially. As mentioned in the previous section, the
maximum wave height depends on the number of waves in
the wave train in both linear and nonlinear theory as shown
in equation (11). Hence the large number of waves is
fundamental for the convergence both of the tail of the
PDF of wave height and the maximum wave height.
Therefore five different realizations with different sets of
random phases have been performed. The duration of each
realization was 32 min. The total number of wave heights
(counting both upcrossing and down-crossing) recorded for
each spectral shape at each probe was about 12800 waves.
This is a sufficient number of waves to check the sensitivity
of the maximum wave height distribution on the number of
waves. In our analysis we have removed the first 200 s of
the records for each realization. This lapse of time was
calculated as the approximate time needed for the wave of
twice the peak frequency to reach the last probe.
4. Comparison of the Laboratory Data and
Theory
4.1. Skewness and Kurtosis of Surface Elevation
[16] In Figure 2 the skewness and kurtosis of the surface
elevation for the experimental data along the channel are
shown. As can be noticed, both skewness and kurtosis
deviate substantially from the Gaussian expected values.
Traditionally, departures from Gaussian statistics have been
attributed to the presence of bound waves; these waves do
not obey the linear dispersion relation and are the result of
random combination of Stokes like waves. The theory, up to
4 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
For the present data, if one takes as m0 the one calculated
from the first probe, we obtain the following results mb3 =
0.22 and mb4 = 3.13. In Figure 2 the horizontal lines
correspond to the expected value from second-order narrow
band theory. According to (20) – (21), a simple relation
between skewness and kurtosis exists: mb4 = 3 + 83 mb2
3 [see
also Marthinsen and Winterstein, 1992; Mori and Janssen,
2006]; this quadratic relation, with our experimental data is
shown in Figure 3. The empirical relation by Guedes Soares
et al. [2003] is also shown in Figure 3. As can be seen there
is no agreement (except for the point that corresponds to the
first probe closest to the wave generator), although both the
theoretical and empirical curve show qualitatively a similar
tendency. As expected most of experimental data of
skewness and kurtosis are underpredicted along the tank if
only second-order theory is considered. This is especially
true for the kurtosis; as was mentioned before, while the
kurtosis can be notably influenced by the dynamics of the
free modes of four wave interactions, the skewness is only
weakly affected. A much better agreement between
experimental data and theory can be achieved if the
dynamics of free waves is included. In Figure 4 we show
a comparison of the experimental data with theory, i.e.,
equation (2). Even though the agreement is not impressive,
the theory seems to reproduce the behavior of the
experimental data in a much better way than the secondorder theory.
4.2. Wave Height Distribution
[17] In this section we will compare the experimental wave
height distribution with results obtained using equation (8).
Hereafter, for brevity, we will refer to equation (8) as
the MER (Modified Edgeworth Rayleigh) distribution.
Note that the use of the MER distribution implies that the
kurtosis of the surface elevation is known. This can be
estimated, either by theory or directly from the experimental
data; the second choice has been adopted. The comparison
of the experimental exceedance probability of wave heights
Figure 2.
(a) Skewness and (b) kurtosis of the
experimental data along the tank as a function of the
distance from the wave maker. l corresponds to a
characteristic wavelength corresponding to the peak frequency period 1.5 s of the waves at the first probe.
second order, has been developed by Longuet-Higgins
[1963]. In order to give here a first estimation of the
skewness and kurtosis, we will adopt the narrow band
approximation of the second-order theory. Under this condition the skewness, mb3 and kurtosis mb4 take the following
form:
1=2
mb3 ¼ 3k0 m0
ð20Þ
mb4 ¼ 3 þ 24k02 m0
ð21Þ
Figure 3. Relationship between skewness and kurtosis.
The solid circles are the experimental data, the solid line is
the second-order nonlinear theory by Marthinsen and
Winterstein [1992], and the dashed line is the emperical
fit by Guedes Soares et al. [2003].
5 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
Figure 4. Kurtosis of the experimental data as a function
of the distance from the wave maker compared with
theoretical values obtained from quasi-resonant theory.
compared with the MER distribution is shown in Figure 5
for a number of probes located at different distances
from the wave maker. Figure 5a corresponds to probe 1 in
Figure 2 and is located 10 m offshore from the wave maker.
Here we emphasize that waves have been generated by
using the random phase approximation so that we expect
that, after a few wavelengths, the surface elevation is still
very close to be characterized by Gaussian distribution (the
experimental value of the kurtosis is m4 = 3.06, i.e., almost
Gaussian). As previously mentioned, as m4 tends to 3.0, the
MER distribution tends to the Rayleigh distribution. Note
that both these theoretical curves in Figure 5a overestimate
the experimental results. The reason of this discrepancy may
be attributed to the fact that experimental data are characterized by a spectrum that has some finite width. Similar
results can be obtained by linear random simulations [i.e.,
Goda, 2000]. As waves travel along the tank, quasi resonant
four wave interactions become important and, as previously
described, the kurtosis of the surface elevation starts to
deviate from the Gaussian value and the tail of the distribution deviates from linear expectation. This is shown in
Figures 5b and 5c obtained from time series recorded
respectively at about 8 and 34 wavelengths from the wave
maker. Because of the aforementioned nonlinear effects, the
exceedance probability obtained from the laboratory data
shows some deviations for large waves from the Rayleigh
distribution. The MER distribution follows this behavior;
this is especially true for waves with large kurtosis as shown
in Figure 5c (m4 = 4.10). It should be noted that the MER
distribution includes only the kurtosis and does not consider
the skewness, which is mainly modified by the presence of
bound modes. As already denoted in the previous section,
the effect of bound modes is negligible if the wave height
distribution for a narrow-band wave train is considered; the
second-order corrections are relevant only for the crest
distribution.
[18] In Figure 6 we show the probability of occurrence of
freak waves Hmax/hrms 8 for short time records, as a
function of the kurtosis measured at different probes. We
Figure 5. Comparison of exceedance wave height
distribution H/hrms: circles, experimental data; solid line,
MER distribution/equation (8); and dashed line, Rayleigh
distribution.)
6 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
series containing each 150 waves; the maximum wave
heights are then collected from each shorter time series. In
Figure 7a the comparison between theory and experiment is
shown for probe 1. The peak of the observed maximum
Figure 6. Occurrence probability of freak wave in the long
recorded wave data N = 11900: circles, experimental data;
solid line, MER distribution by equation (19); and dashed
line, Rayleigh distribution.
assumed 100 number of waves for each record and counted
number of freak waves. The solid circles denote the experimental data and the linear theory based on the Rayleigh
distribution is represented by the dotted line, the nonlinear
theory, equation (19), corresponds to the solid line. The
occurrence probability of freak waves in the experimental
data shows a clear dependence on the kurtosis (this effect of
course is not described by the Rayleigh distribution). While
the occurrence probability of freak waves estimated by the
MER distribution shows a linear relation with the kurtosis,
equation (19), experimental data appear to give a quadratic
function of the kurtosis. This is probably because the MER
includes only the lowest correction of nonlinearity and
excludes high-order cumulants. Moreover, the MER seems
to overestimate the occurrence probability of freak wave
with respect to the experimental data for small kurtosis. This
may be related to the fact that the MER distribution (as well
as the Rayleigh distribution) is based on the assumption that
waves are narrow banded.
4.3. Maximum Wave Height Distribution
[19] The consequences of the maximum wave height
distribution (equation (11)) are in general hard to verify,
not only because equation (11) depends on the number of
waves considered but also because nonlinear effects are
included in the distribution via the kurtosis which requires a
large number of data to converge. Nevertheless, the present
experimental data set seems to be suitable for such comparison (as previously described, the time series are very
long). Figure 7 shows the comparison of the maximum
wave height distribution from the experimental data with
equation (11) for N = 150. The solid circles represent the
experimental data, the maximum wave height distribution
from Rayleigh theory (denoted Rayleigh Hmax distribution
hereafter) is represented by the dotted line and equation (12)
(denoted by MER Hmax distribution hereafter) corresponds
to the solid line. To obtain a distribution of maximum wave
height, each experimental record is divided in smaller time
Figure 7. Comparison of maximum wave height distribution Hmax/hrms with N = 150: circles, experimental data;
solid line, MER Hmax distribution; and dashed line,
Rayleigh Hmax distribution.
7 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
C09011
of finite width of the spectrum). As waves propagate
through the flume, the nonlinear dynamics result in an
increase of the kurtosis, and therefore the maximum wave
height distribution of the laboratory data departs from the
Rayleigh distribution (Figure 7a – 7c) and the peak of the
observed distribution shift to larger wave heights. While
the Rayleigh Hmax distribution is independent of the kurtosis, the MER Hmax distribution follows decently the behavior of the experimental data. Figure 8 shows similar
comparisons with N = 250. The peak of the experimental
data in the linear wave condition, Figure 8a, moves from H/
hrms ’ 6.2 to H/hrms ’ 6.6 because of the change of number
of waves in the wave train from 150 to 250. The MER Hmax
distribution seems to reproduce reasonably well the experimental data; for very small kurtosis, it overestimates the
experimental data and for large kurtosis it slightly underestimates the data. This is consistent with what we have
shown in Figure 6 where the occurrence probability of freak
waves is plotted as a function of the kurtosis. Note that
according to the observations and theory more than half of
the number of waves in the case of m4 = 4.1 and N = 250
satisfy the freak wave condition as shown in Figure 8c.
[20] Here we also discuss the general behavior of the
probability density function of maximum wave height in the
Figure 8. Comparison of maximum wave height distribution Hmax/hrms with N = 250: circle, experimental data; solid
line, MER Hmax distribution; and dashed line, Rayleigh
Hmax distribution.
wave height distribution is larger than the Rayleigh Hmax
distribution and the MER Hmax distribution with m4 = 3.06
but the observed distribution is more narrow (just as for the
wave height distribution we ascribe this difference to effects
Figure 9. Comparison of expected value of Hmax/H1=3,
hHmax/H1=3i.
8 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
Figure 10. Comparison of occurrence probability of freak
wave as a function of m4 and N.
nonlinear wavefield, by plotting the expected value of the
maximum wave height, indicated by hi, as a function of m4
and N, see Figure 9. In this figure we show the numerically
integrated value of equation (11) and the ensemble average
of the experimental data. The hHmax/H1/3i according to the
Rayleigh theory corresponds to Figure 9 with m4 = 3. The
dependence of hHmax/H1/3i on m4 and N is clear in both
the experimental data and equation (11). Overall, hHmax/H1/3i
of the experimental data is smaller than the MER Hmax
distribution; nevertheless, theoretical and experimental data
show similar trends.
[21] Finally, Figure 10 shows the comparison between
the experimental data and theory of freak wave occurrence
frequency, Pfreak. For small kurtosis waves, m4 < 3.5, the
freak wave occurrence frequency estimated by the MER
Hmax distribution is larger than what is obtained from the
experimental data. This tendency is similar to Figure 6.
However, the experimental data show a rapid growth of
the occurrence probability of freak waves for larger
kurtosis, m4 > 3.5.
5. Conclusion
[22] Four wave quasi-resonant interactions play an important role in the determination of the statistical properties
C09011
of the surface elevation. This discovery, which is fairly new,
is of some relevance in the community of extreme wave
forecasting. Previously, deviations from Gaussian behavior
have been attributed only to bound modes. However, even
for the most severe sea state conditions, the value of the
kurtosis from second-order theory barely reaches values
larger than 3.15. Moreover, the contribution to wave height
distribution from second-order theory is practically negligible (in the narrow-band approximation it is exactly zero). If
waves are long crested and sufficiently steep, the dynamics
of the free waves can bring very strong departures from
Gaussian expectation. In the experimental data analyzed in
the present paper, values as high as 4.1 have been measured.
Large values of kurtosis can substantially change the tails of
the probability density function of wave height.
[23] In the present paper a detailed comparison between
experimental data, consisting of a large number of waves,
and the theoretical expectation described by Mori and
Janssen [2006] have been discussed. The spatial evolution
of kurtosis along the tank can be estimated by the quasiresonant four wave interactions theory originally developed
by Janssen [2003]. As the kurtosis increases, the probability
of large amplitude waves also increases. The MER distribution is capable of describing with some accuracy the tail
of the exceedance probability of wave heights for large
kurtosis, i.e., for strong nonlinearity. Comparison of the
MER distribution for maximum wave height with experimental data has also shown a much better agreement than
when the maximum wave height distribution obtained from
Rayleigh distribution is used. Overall, the theory originally
described by Edgeworth at the beginning of the last century,
combined with four wave quasi-resonant interaction theory
and wave statistical theory seems to be an interesting
approach for predicting extreme waves in long-crested
conditions. We should emphasize here that the theory does
not need any empirical, ad hoc parameter.
[24] In real sea states directional effects are important. In
this paper we have concentrated on the idea of extreme
wave generation owing to four wave interactions in a
unidirectional wave train. Presently we are extending the
theory to including directional wave effects. This will be
discussed in near future.
[25] Acknowledgments. NM and MO wish to thank the European
Centre for Medium-Range Weather Forecasts FOR financial support during
their stay as visiting scientists there. NM also wishes to thank JSPS for
financial support. We thank the European Union (contract HPRI-CT-200100176) for making the experiments in Marintek possible. C. T. Stansberg
and L. Cavaleri are also acknowledged for their valuable help during the
experiments.
References
Annekov, S., and V. Shrira (2006), Role of non-resonant interactions in
evolution of nonlinear random water wave fields, J. Fluid Mech., 561,
181 – 207.
Boccotti, P. (1989), On mechanics of irregular gravity waves, Atti Accad.
Naz. Lincei Mem., 19, 11 – 170.
Edgeworth, F. (1907), On the representation of statistical frequency by a
series, J. R. Stat. Soc., 70, 102 – 106.
Fedele, F., and F. Arena (2005), Weakly nonlinear statistics of high nonlinear random waves, Phys. Fluids., 17, 026601, doi:10.1063/1.1831311.
Fedele, F., and A. Tayfun (2006), Explaining freak waves by a stochastic
theory of wave groups, paper presented at 25th International Conference
on Offshore Mechanics and Arctic Engineering, Am. Soc. of Mech. Eng.,
Hamburg, Germany.
9 of 10
C09011
MORI ET AL.: EXTREME STATISTICS OF DEEP WATER WAVES
Goda, Y. (2000), Random Seas and Design of Maritime Structures, 2nd ed.,
World Sci., Hackensack, N. J.
Guedes Soares, C., Z. Cherneva, and E. Antão (2003), Characteristics of
abnormal waves in North Sea storm sea states, Appl. Ocean Res., 25,
337 – 344.
Guedes Soares, C., Z. Cherneva, and E. M. Antã o (2004), Abnormal waves
during Hurricane Camille, J. Geophys. Res., 109, C08008, doi:10.1029/
2003JC002244.
Hasselmann, K. (1962), On the nonlinear energy transfer in gravity-wave
spectrum. I. Gerenral theory, Jo. Fluid Mech., 12, 481 – 500.
Janssen, P. (2003), Nonlinear four-wave interactions and freak waves,
J. Phys. Oceanogr., 33, 863 – 884.
Krasitskii, V. (1994), On reduced equations in the Hamiltonian theory of
weakly nonlinear surface waves, J. Fluid Mech., 272, 1 – 20.
Longuet-Higgins, M. (1952), On the statistical distribution of the heights of
sea waves, J. Mar. Res., 11, 245 – 266.
Longuet-Higgins, M. (1963), The effect of non-linearities on statistical
distributions in the theory of sea waves, J. Fluid Mech., 17, 459 – 480.
Marthinsen, T., and S. Winterstein (1992), On the skewness of random
surface waves, in Proceedings of the 2nd International Offshore and
Polar Engineering Conference, vol. 3, pp. 472 – 478, Int. Soc. of Offshore and Polar Eng., San Francisco, Calif.
Mori, N., and P. Janssen (2006), On kurtosis and occurrence probability of
freak waves, J. Phys. Oceanogr., 36, 1471 – 1483.
Mori, N., and T. Yasuda (2002), A weakly non-Gaussian model of wave
height distribution for random wave train, Ocean Eng., 29, 1219 – 1231.
Mori, N., T. Yasuda, and K. Kawaguchi (1997), Breaking effects on waves
statistics for deep-water random wave train, in Ocean Wave Measurement
and Analysis (1997), edited by B. L. Edge and J. M. Hemsley, pp. 316 –
328, Am. Soc. of Civ. Eng., Reston, Va.
Mori, N., P. Liu, and T. Yasuda (2002), Analysis of freak wave measurements in the Sea of Japan, Ocean Eng., 29, 1399 – 1414.
Muller, P., and D. Henderson (Eds.) (2005), Proceedings of Rogue Waves,
Hawaiian Winter Workshop, Univ. of Hawai’i at Manoa, Honolulu.
Næss, A. (1985), On the distribution of crest-to-trough wave heights,
Ocean Eng., 12, 221 – 234.
Olagnon, M., , and M. Prevosto (Eds.) (2004), Proceedings of Rogue Waves
2004, Inst. Fr. de Rech. Pour l’Exploit. de la Mer, Brest.
Onorato, M., A. Osborne, M. Serio, and S. Bertone (2001), Freak waves in
random oceanic sea states, Phys. Rev. Lett., 86, 5831 – 5834.
Onorato, M., A. Osborne, and M. Serio (2002), Extreme wave events in
directional, random oceanic sea states, Phys. Fluids, 14, L25 – L28.
Onorato, M., A. Osborne, M. Serio, M. Brandini, and C. Stansberg (2004),
Observation of strongly non-Gaussian statistics for random sea surface
gravity waves in wave flume experiments, Phys. Rev. E, 70, 067302,
doi:10.1103/PhysRevE.70.067302.
Onorato, M., A. Osborne, and M.Serio (2005a), On deviations from
Guassian statistics for surface gravity waves, in Proceedings Rogue
C09011
Waves, Hawaiian Winter Workshop, edited by P. Muller and D. Henderson,
pp. 79 – 83, Univ. of Hawai’i at Manoa, Honolulu.
Onorato, M., A. Osborne, M. Serio, and L. Cavaleri (2005b), Modulational
instability and non-Gaussian statistics in experimental random waterwave trains, Phys. Fluids, 17, 078101, doi:10.1063/1.1946769.
Onorato, M., A. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. Stansberg
(2006), Extreme waves, modulational instability and second order theory:
Wave flume experiments on irregular waves, Eur. J. Mech. B, 25, 586 –
601.
Pelinovsky, E., and C. Kharif (2006), Rogue waves Vienna, Austria, Eur. J.
Mech. B, 25, 535 – 692.
Socquet-Juglard, H., K. Dysthe, K. Trulsen, H. E. Krogstad, and J. Liu
(2005), Distribution of surface gravity waves during spectral changes,
J. Fluid Mech., 542, 195 – 216.
Stansell, P., J. Wolfram, and S. Zachary (2003), Horizontal asymmetry and
steepness distributions for wind-driven ocean waves, Appl. Ocean Res.,
25, 137 – 155.
Tayfun, M. (1980), Narrow band nonlinear sea waves, J. Geophys. Res., 85,
1548 – 1552.
Tayfun, M. (1981), Distribution of crest-to-trough wave heights, J. Waterw.
Port Coastal Ocean Eng., 107, 149 – 158.
Tayfun, M. (2006), Statistics of nonlinear wave crests and groups, Ocean
Eng., 33, 589 – 1622.
Tayfun, M., and J.-M. Lo (1990), Nonlinear effects on wave envelope and
phase, J. Waterw. Port Coastal Ocean Eng., 116, 79 – 100.
Yasuda, T., and N. Mori (1994), High order nonlinear effects on deep-water
random wave trains, in Waves—Physical and Numerical Modelling, vol. 2,
edited by M. Isaacson and M. C. Quick, pp. 823 – 332, Vancouver, B. C.,
Canada.
Yasuda, T., and N. Mori (1997), Occurrence properties of giant freak waves
the sea area around Japan, J. Waterw. Port Coastal Ocean Eng., 123,
209 – 213.
Yasuda, T., N. Mori, and K. Ito (1992), Freak waves in a unidirectional
wave train and their kinematics, in Coastal Engineering (1992), vol. 1,
edited by B. L. Edge, pp. 751 – 764, Am. Soc. of Civ. Eng., Reston, Va.
Yasuda, T., N. Mori, and S. Nakayama (1997), Characteristics of giant freak
waves observed in the Sea of Japan, in Ocean Wave Measurement and
Analysis (1997), edited by B. L. Edge and J. M. Hemsley, pp. 482 – 495,
Am. Soc. of Civ. Eng., Reston, Va.
P. A. E. M. Janssen, European Centre for Medium-Range Weather
Forecasts, Shinfield Park, Reading RG2 9AX, UK. (peter.janssen@ecmwf.
int)
N. Mori, Graduate School of Engineering, Osaka City University, Osaka
558-8585, Japan. ([email protected])
M. Onorato, A. R. Osborne, and M. Serio, Dipartmento di Fisica
Generale, Università di Torino, Via P. Giardia, I-10125 1-Torino, Italy.
([email protected]; [email protected]; [email protected])
10 of 10